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Order-sorted logic: Fix CS1 deprecated date parameter errors

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{{Quantum field theory|cTopic=Tools}}
'''Wick's theorem''' is a method of reducing high-[[Differential equation#Types of differential equations|order]] [[derivative]]s to a [[combinatorics]] problem.<ref>Philips, 2001</ref> It is named after [[Gian-Carlo Wick]]. It is used extensively in [[Quantum Field Theory|quantum field theory]] to reduce arbitrary products of [[creation and annihilation operators]] to sums of products of pairs of these operators. This allows for the use of [[Green's function (many-body theory)|Green's function methods]], and consequently the use of [[Feynman diagram]]s in the field under study. A more general idea in [[probability theory]] is [[Isserlis’ theorem]].

==Definition of contraction==
For two operators <math>\hat{A}</math> and <math>\hat{B}</math> we define their contraction to be
:<math>\hat{A}^\bullet\, \hat{B}^\bullet \equiv \hat{A}\,\hat{B}\, - \mathopen{:} \hat{A}\,\hat{B} \mathclose{:}</math>

where <math>\mathopen{:} \hat{O} \mathclose{:}</math> denotes the [[normal order]] of an operator <math>\hat{O}</math>.

Alternatively, contractions can be denoted by a line joining <math>\hat{A}</math> and <math>\hat{B}</math>.

We shall look in detail at four special cases where <math>\hat{A}</math> and <math>\hat{B}</math> are equal to creation and annihilation operators. For <math>N</math> particles we'll denote the creation operators by <math>\hat{a}_i^\dagger</math> and the annihilation operators by <math>\hat{a}_i</math> (<math>i=1,2,3\ldots,N</math>).
They satisfy the usual commutation relations <math>[\hat{a}_i,\hat{a}_j^\dagger]=\delta_{ij}</math>, where <math>\delta_{ij}</math> denotes the [[Kronecker delta]].

We then have

:<math>\hat{a}_i^\bullet \,\hat{a}_j^\bullet = \hat{a}_i \,\hat{a}_j \,- \mathopen{:}\,\hat{a}_i\, \hat{a}_j\,\mathclose{:}\, = 0</math>
:<math>\hat{a}_i^{\dagger\bullet}\, \hat{a}_j^{\dagger\bullet} = \hat{a}_i^\dagger\, \hat{a}_j^\dagger \,-\,\mathopen{:}\hat{a}_i^\dagger\,\hat{a}_j^\dagger\,\mathclose{:}\, = 0</math>
:<math>\hat{a}_i^{\dagger\bullet}\, \hat{a}_j^\bullet = \hat{a}_i^\dagger\, \hat{a}_j \,- \mathopen{:}\,\hat{a}_i^\dagger \,\hat{a}_j\, \mathclose{:}\,= 0</math>
:<math>\hat{a}_i^\bullet \,\hat{a}_j^{\dagger\bullet}= \hat{a}_i\, \hat{a}_j^\dagger \,- \mathopen{:}\,\hat{a}_i\,\hat{a}_j^\dagger \,\mathclose{:}\, = \delta_{ij}</math>
where <math>i,j = 1,\ldots,N</math>.

These relationships hold true for bosonic operators or fermionic operators because of the way normal ordering is defined.

==Examples==

We can use contractions and normal ordering to express any product of creation and annihilation operators as a sum of normal ordered terms. This is the basis of Wick's theorem. Before stating the theorem fully we shall look at some examples.

Suppose <math>\hat{a}_i</math> and <math>\hat{a}_i^\dagger</math> are [[bosonic]] operators satisfying the [[commutation relations]]:
:<math>\left [\hat{a}_i^\dagger, \hat{a}_j^\dagger \right] = 0 </math>
:<math>\left [\hat{a}_i, \hat{a}_j \right] = 0 </math>
:<math>\left [\hat{a}_i, \hat{a}_j^\dagger \right ] = \delta_{ij} </math>
where <math>i,j = 1,\ldots,N</math>, <math>\left[ \hat{A}, \hat{B} \right] \equiv \hat{A} \hat{B} - \hat{B} \hat{A}</math> denotes the [[commutator]], and <math>\delta_{ij}</math> is the Kronecker delta.

We can use these relations, and the above definition of contraction, to express products of <math>\hat{a}_i</math> and <math>\hat{a}_i^\dagger</math> in other ways.

=== Example 1 ===

:<math>\hat{a}_i \,\hat{a}_j^\dagger = \hat{a}_j^\dagger \,\hat{a}_i + \delta_{ij} = \hat{a}_j^\dagger \,\hat{a}_i + \hat{a}_i^\bullet \,\hat{a}_j^{\dagger\bullet} =\,\mathopen{:}\,\hat{a}_i\, \hat{a}_j^\dagger \,\mathclose{:} + \hat{a}_i^\bullet \,\hat{a}_j^{\dagger\bullet} </math>

Note that we have not changed <math>\hat{a}_i \,\hat{a}_j^\dagger</math> but merely re-expressed it in another form as <math>\,\mathopen{:}\,\hat{a}_i\, \hat{a}_j^\dagger \,\mathclose{:} + \hat{a}_i^\bullet \,\hat{a}_j^{\dagger\bullet} </math>

=== Example 2 ===

:<math>\hat{a}_i \,\hat{a}_j^\dagger \, \hat{a}_k= (\hat{a}_j^\dagger \,\hat{a}_i + \delta_{ij})\hat{a}_k = \hat{a}_j^\dagger \,\hat{a}_i\, \hat{a}_k + \delta_{ij}\hat{a}_k = \hat{a}_j^\dagger \,\hat{a}_i\,\hat{a}_k + \hat{a}_i^\bullet \,\hat{a}_j^{\dagger\bullet} \hat{a}_k =\,\mathopen{:}\,\hat{a}_i\, \hat{a}_j^\dagger \hat{a}_k \,\mathclose{:} + \mathopen{:}\,\hat{a}_i^\bullet \,\hat{a}_j^{\dagger\bullet} \,\hat{a}_k \mathclose{:} </math>

=== Example 3 ===

:<math>\hat{a}_i \,\hat{a}_j^\dagger \, \hat{a}_k \,\hat{a}_l^\dagger= (\hat{a}_j^\dagger \,\hat{a}_i + \delta_{ij})(\hat{a}_l^\dagger\,\hat{a}_k + \delta_{kl})</math>
:::::<math> = \hat{a}_j^\dagger \,\hat{a}_i\, \hat{a}_l^\dagger\, \hat{a}_k + \delta_{kl}\hat{a}_j^\dagger \,\hat{a}_i + \delta_{ij}\hat{a}_l^\dagger\hat{a}_k + \delta_{ij} \delta_{kl} </math>
:::::<math> = \hat{a}_j^\dagger (\hat{a}_l^\dagger\, \hat{a}_i + \delta_{il}) \hat{a}_k + \delta_{kl}\hat{a}_j^\dagger \,\hat{a}_i + \delta_{ij}\hat{a}_l^\dagger\hat{a}_k + \delta_{ij} \delta_{kl} </math>
:::::<math>= \hat{a}_j^\dagger \hat{a}_l^\dagger\, \hat{a}_i \hat{a}_k + \delta_{il} \hat{a}_j^\dagger \, \hat{a}_k + \delta_{kl}\hat{a}_j^\dagger \,\hat{a}_i + \delta_{ij}\hat{a}_l^\dagger\hat{a}_k + \delta_{ij} \delta_{kl} </math>
:::::<math>= \,\mathopen{:}\hat{a}_i \,\hat{a}_j^\dagger \, \hat{a}_k \,\hat{a}_l^\dagger\,\mathclose{:} + \mathopen{:}\,\hat{a}_i^\bullet \,\hat{a}_j^\dagger \, \hat{a}_k \,\hat{a}_l^{\dagger\bullet}\,\mathclose{:}+\mathopen{:}\,\hat{a}_i \,\hat{a}_j^\dagger \, \hat{a}_k^\bullet \,\hat{a}_l^{\dagger\bullet}\,\mathclose{:}+\mathopen{:}\,\hat{a}_i^\bullet \,\hat{a}_j^{\dagger\bullet} \, \hat{a}_k \,\hat{a}_l^\dagger\,\mathclose{:}+ \,\mathopen{:}\hat{a}_i^\bullet \,\hat{a}_j^{\dagger\bullet} \, \hat{a}_k^{\bullet\bullet}\,\hat{a}_l^{\dagger\bullet\bullet} \mathclose{:} </math>

In the last line we have used different numbers of <math>^\bullet</math> symbols to denote different contractions. By repeatedly applying the commutation relations it takes a lot of work, as you can see, to express <math>\hat{a}_i \,\hat{a}_j^\dagger \, \hat{a}_k \,\hat{a}_l^\dagger</math> in the form of a sum of normally ordered products. It is an even lengthier calculation for more complicated products.

Luckily Wick's theorem provides a shortcut.

== Statement of the theorem ==

A product of creation and annihilation operators <math>\hat{A} \hat{B} \hat{C} \hat{D} \hat{E} \hat{F}\ldots </math> can be expressed as
:<math>
\begin{align}
\hat{A} \hat{B} \hat{C} \hat{D} \hat{E} \hat{F}\ldots &= \mathopen{:} \hat{A} \hat{B} \hat{C} \hat{D} \hat{E} \hat{F}\ldots \mathclose{:} \\
&\quad + \sum_\text{singles} \mathopen{:} \hat{A}^\bullet \hat{B}^\bullet \hat{C} \hat{D} \hat{E} \hat{F} \ldots \mathclose{:} \\
&\quad + \sum_\text{doubles} \mathopen{:} \hat{A}^\bullet \hat{B}^{\bullet\bullet} \hat{C}^{\bullet\bullet} \hat{D}^\bullet \hat{E} \hat{F} \ldots \mathclose{:} \\
&\quad + \ldots
\end{align}
</math>

In other words, a string of creation and annihilation operators can be rewritten as the normal-ordered product of the string, plus the normal-ordered product after all single contractions among operator pairs, plus all double contractions, etc., plus all full contractions.

Applying the theorem to the above examples provides a much quicker method to arrive at the final expressions.

'''A warning''': In terms on the right hand side containing multiple contractions care must be taken when the operators are fermionic. In this case an appropriate minus sign must be introduced according to the following rule: rearrange the operators (introducing minus signs whenever the order of two fermionic operators is swapped) to ensure the contracted terms are adjacent in the string. The contraction can then be applied (See Rule C″ in Wick's paper).

Example:

If we have two fermions (<math>N=2</math>) with creation and annihilation operators <math>\hat{f}_i^\dagger</math> and <math>\hat{f}_i</math> (<math>i=1,2</math>) then

:<math> \begin{array}{ll} \hat{f}_1 \,\hat{f}_2 \, \hat{f}_1^\dagger \,\hat{f}_2^\dagger \,&= \,\mathopen{:} \hat{f}_1 \,\hat{f}_2 \, \hat{f}_1^\dagger \,\hat{f}_2^\dagger \, \mathclose{:} \\ & - \,\mathopen{:} \hat{f}_1^\bullet \,\hat{f}_2 \, \hat{f}_1^{\dagger\bullet} \,\hat{f}_2^\dagger \, \mathclose{:} + \,\mathopen{:} \hat{f}_1^\bullet \,\hat{f}_2 \, \hat{f}_1^\dagger \,\hat{f}_2^{\dagger\bullet} \, \mathclose{:} +\,\mathopen{:} \hat{f}_1 \,\hat{f}_2^\bullet \, \hat{f}_1^{\dagger\bullet} \,\hat{f}_2^\dagger \, \mathclose{:} - \mathopen{:} \hat{f}_1 \,\hat{f}_2^\bullet \, \hat{f}_1^\dagger \,\hat{f}_2^{\dagger\bullet} \, \mathclose{:} \\ & -\mathopen{:} \hat{f}_1^{\bullet\bullet} \,\hat{f}_2^\bullet \, \hat{f}_1^{\dagger\bullet\bullet} \,\hat{f}_2^{\dagger\bullet} \, \mathclose{:}+\mathopen{:} \hat{f}_1^{\bullet\bullet} \,\hat{f}_2^\bullet \, \hat{f}_1^{\dagger\bullet} \,\hat{f}_2^{\dagger\bullet\bullet}\mathclose{:} \end{array} </math>

Note that the term with contractions of the two creation operators and of the two annihilation operators is not included because their contractions vanish.

==Wick's theorem applied to fields==

:<math>\mathcal C(x_1, x_2)=\left \langle 0 |\mathcal T\phi_i(x_1)\phi_i(x_2)|0\right \rangle=\overline{\phi_i(x_1)\phi_i(x_2)}=i\Delta_F(x_1-x_2)
=i\int{\frac{d^4k}{(2\pi)^4}\frac{e^{-ik(x_1-x_2)}}{(k^2-m^2)+i\epsilon}}.</math>

Which means that <math>\overline{AB}=\mathcal TAB-\mathopen{:}AB\mathclose{:}</math>

In the end, we arrive at Wick's theorem:

The T-product of a time-ordered free fields string can be expressed in the following manner:

:<math>\mathcal T\Pi_{k=1}^m\phi(x_k)=\mathopen{:}\Pi\phi_i(x_k)\mathclose{:}+\sum_{\alpha,\beta}\overline{\phi(x_\alpha)\phi(x_\beta)}\mathopen{:}\Pi_{k\not=\alpha,\beta}\phi_i(x_k)\mathclose{:}+
</math>

:<math>\mathcal
+\sum_{(\alpha,\beta),(\gamma,\delta)}\overline{\phi(x_\alpha)\phi(x_\beta)}\;\overline{\phi(x_\gamma)\phi(x_\delta)}\mathopen{:}\Pi_{k\not=\alpha,\beta,\gamma,\delta}\phi_i(x_k)\mathclose{:}+\cdots.
</math>

Applying this theorem to [[s matrix|S-matrix]] elements, we discover that normal-ordered terms acting on [[vacuum state]] give a null contribution to the sum. We conclude that ''m'' is even and only completely contracted terms remain.

:<math>F_m^i(x)=\left \langle 0 |\mathcal T\phi_i(x_1)\phi_i(x_2)|0\right \rangle=\sum_\mathrm{pairs}\overline{\phi(x_1)\phi(x_2)}\cdots
\overline{\phi(x_{m-1})\phi(x_m})</math>

:<math>G_p^{(n)}=\left \langle 0 |\mathcal T\mathopen{:}v_i(y_1)\mathclose{:}\dots\mathopen{:}v_i(y_n)\mathclose{:}\phi_i(x_1)\cdots \phi_i(x_p)|0\right \rangle</math>

where ''p'' is the number of interaction fields (or, equivalently, the number of interacting particles) and ''n'' is the development order (or the number of vertices of interaction). For example, if <math>v=gy^4 \Rightarrow \mathopen{:}v_i(y_1)\mathclose{:}=\mathopen{:}\phi_i(y_1)\phi_i(y_1)\phi_i(y_1)\phi_i(y_1)\mathclose{:}</math>

This is analogous to the [[Multinormal distribution#Higher moments|corresponding theorem]] in statistics for the [[moment (mathematics)|moments]] of a [[Gaussian distribution]].

Note that this discussion is in terms of the usual definition of normal ordering which is appropriate for the [[vacuum expectation value]]s (VEV's) of fields. (Wick's theorem provides as a way of expressing VEV's of ''n'' fields in terms of VEV's of two fields.<ref>See for example also: Mrinal Dasgupta: [http://www.hep.manchester.ac.uk/u/dasgupta/teaching/Dasgupta-08-Intro-to-QFT.pdf An introduction to Quantum Field Theory], Lectures presented at the RAL School for High Energy Physics, Somerville College, Oxford, September 2008, section 5.1 Wick's Theorem (downloaded 3 December 2012)</ref>) There are any other possible definitions of normal ordering, and Wick's theorem is valid irrespective. However Wick's theorem only simplifies computations if the definition of normal ordering used is changed to match the type of expectation value wanted. That is we always want the expectation value of the normal ordered product to be zero. For instance in
[[Thermal quantum field theory|thermal field theory]] a different type of expectation value, a thermal trace over the density matrix, requires a different definition of [[Normal_ordering#Alternative_definitions|normal ordering]].<ref>(Evans and Steer, 1996)</ref>

== References ==

<references />
* [[Gian-Carlo Wick|G.C. Wick]], [http://link.aps.org/abstract/PR/v80/p268 The Evaluation of the Collision Matrix], Phys. Rev. 80, 268 - 272 (1950)
* Silvan S. Schweber, An Introduction to Relativistic Quantum Field Theory, Harper and Row, New York (1962). (Chapter 13, Sec c)
* [[Michael Peskin|M. E. Peskin]] and [[Daniel V. Schroeder|D. V. Schroeder]], An Introduction to Quantum Field Theory, Perseus Books (1995). (§4.3)
*{{cite web |url=http://www.math.sunysb.edu/~tony/whatsnew/column/feynman-1101/feynman1.html |title=Finite-dimensional Feynman Diagrams |accessdate=2007-10-23 |author=Tony Philips |year=2001 |month=11 |work=What's New In Math |publisher=[[American Mathematical Society]]}}
*{{cite web |url=http://www.ua.es/cuantica/docencia/otros/cc/node15.html |title=Wick's theorem |accessdate=2008-07-29 |author=Emilio San Fabian |year=2001 |month=2}}
* T.S. Evans, D.A. Steer, [http://arxiv.org/abs/hep-ph/9601268 Wick's theorem at finite temperature], Nucl. Phys B 474, 481-496 (1996) [http://arxiv.org/abs/hep-ph/9601268 arXiv:hep-ph/9601268]

[[Category:Quantum field theory]]
[[Category:Scattering theory]]
[[Category:Physics theorems]]

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en>Monkbot: /* Order-sorted logic */Fix CS1 deprecated date parameter errors